I'm not really sure which is best, or how they're even different?
I think they will offer a similar viewing experience. However, I will say you will probably get slightly better resolution on a computer monitor compared to the TV monitor, which isn't really a big deal unless you are planning on sitting really close to the screen. In this case you might be able to see the individual pixels on the TV monitor easier than on a computer monitor.
I personally prefer gaming on monitors. The resolution and color stand out to me more than gaming on TVs. Don't get me wrong, though. There's nothing like playing a game on a 65 inch plasma TV.
It really depends on what kind of performance you're trying to get out of it.
If you want raw resolution, the answer would be a computer monitor. If you're looking for size and versatility, regular HDTVs may work as well. If you are a power user, I would personally recommend getting a 120Hz monitor, as it truly enhances the amount of stuff you can see and overall, rendering and picture quality are smoother.
I use the Asus GD235HZ 24" monitor for my gaming needs. It is a bit pricier than some cheaper, 60Hz screens, but for raw performance, it really is more enjoyable. Also, if you are so inclined, you can use 120Hz monitors as 3D monitors as well.